Herrero, Miguel A. and Velázquez, J.J. L.
(1996)
*Singularity patterns in a chemotaxis model.*
Mathematische Annalen, 306
(1).
pp. 583-623.
ISSN 0025-5831

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## Abstract

The authors study a chemotactic model under certain assumptions and obtain the existence of a class of solutions which blow up at the center of an open disc in finite time. Such a finite-time blow-up of solutions implies chemotactic collapse, namely, concentration of species to form sporae. The model studied is the limiting case of a basic chemotactic model when diffusion of the chemical approaches infinity, which has the form ut=Δu−χ(uv), 0=Δv+(u−1), on ΩR2, where Ω is an open disc with no-flux (homogeneous Neumann) boundary conditions. The initial conditions are continuous functions u(x,0)=u0(x)≥0, v(x,0)=v0(x)≥0 for xΩ. Under these conditions, the authors prove there exists a radially symmetric solution u(r,t) which blows up at r=0, t=T<∞. A specific description of such a solution is presented. The authors also discuss the strong similarity between the chemotactic model they study and the classical Stefan problem.

Item Type: | Article |
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Uncontrolled Keywords: | Blow-up; equations; radial solutions; chemotactic collapse |

Subjects: | Medical sciences > Biology > Biomathematics Sciences > Mathematics > Differential equations |

ID Code: | 17199 |

References: | W. Alt: Orientation of cells migrating in a chemotatic gradient. Lectures Notes in Biomath. vol. 38, Springer-Verlag (1980), 353-366. W. Alt: Biased random walk models for chemotaxis and related diffusion approximations. J. Math. Biol. 9 (1980), 147-177. SB. Angenent, JJ.L. Velázquez: Degenerate neckpinches in mean curvature flow. To appear. E. Bombieri, E. De Giorgi, E. Giusti: Minimal cones and the Bernstein problem. Inventiones Math. 7 (1969), 243-268. S. Childress: Chemotactic collapse in two dimensions. Lectures Notes in Biomath. vol. 55, Springer-Verlag (1984), 61-66. S. Childress, JK. Percus: Nonlinear aspects of chemotaxis. Math. Biosci. 56 (1981), 217-237. J.I. Diaz, T. Nagai: Symmetrization in a parabolic-elliptic system related to chemotaxis. Adv. Math. Sci. Appl., to appear. Y. Giga, RV. Kohn: Asymptolically self-similar blow-up of semilinear heal equations. Comm. Pure Appl. Math. 38(1985),297-319. MA. Herrero, J JL. Velazquez: Explosion des solutions d'équations paraboliques semilinéaires supercritiques. C.R. Acad. Sci. Paris t. 319, 1 (1994), 141-145. MA. Herrero, J JL. Velazquez: On the melting of ice balls. SIAM J Math. Anal, to appear. W. Jäger, S. Luckhaus: On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Amer. Math. Soc., vol. 329, n. 2 (1992), 819-824. EF. Keller, LA. Segel: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26 (1970), 399-415. T. Nagai: Blow-up of radially symmetrie solutions to a chemotaxis system. To appear in Adv. Math. Sci. Appl. V. Nanjundiah: Chemotaxis, signal relaying and aggregation morphology. J. Theor. Biol. 42 (1973),63-105. JJL. Velazquez: Curvature blow-up in perturbations of minimal cones evolving by mean curvature flow. Annali Scuola Normale Superiore di Pisa, Serie IV, vol. XXI, Fase. 4 (1994),595-628. |

Deposited On: | 26 Nov 2012 09:36 |

Last Modified: | 07 Feb 2014 09:43 |

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